Rounding numbers

Have you ever wondered what to do when you just have too many digits to do a quick calculation? When you don’t need an exact, but an approximate result? Do your decimal numbers have an infinite repeating decimal expansion – like when you try to divide number by and the threes just go on forever? Well, there’s a light at the end of the tunnel – a great method called rounding.

Rounding is a method in which use to reduce the number of digits in order to simplify the calculation or if we do not need that much digits, while trying to keep the value of a number approximately the same as the initial value. The choice of values you want to round a number to depends on the smallest order of magnitude you want to work with – in front of the decimal point (units, tens, hundreds, thousands…) or behind it (tenths, hundredths, etc.). The difference between the exact value and the rounded (or approximate) value is called the round-off error.

The sign we use to note that two values are approximately the same (which means “close enough for our needs”) is “≈”. So if we want to say the number is approximately number , we write .

Rounding methods

There are five common methods of rounding numbers which you can use, depending on your current needs. They are:

rounding up,

rounding down,

rounding towards zero,

rounding away from zero and

rounding to the nearestacceptable value.

To round up is increasing the value of a number to the nearest acceptable larger value. So, if we round up to the nearest ten, any number larger than number and smaller than number will be rounded to number .

To round down means the exact opposite – we decrease the value of a number to the nearest acceptable smaller value. So, if we round down to the nearest ten, any number larger than number and smaller than number will be rounded to number .

To round towards zero means to change the value of a number to the nearest acceptable value that is closer to zero than the number we are currently changing.

To round away from zero is to change the value of a number to the nearest acceptable value that is further away from zero than the number we are changing.

To round the number to the nearest acceptable value is to change the value of a number to the nearest acceptable value, and it doesn’t matter if it is larger or smaller than the number we are changing.

If the difference between the number we are rounding and the nearest acceptable smaller value is lesser than the difference between that number and the nearest acceptable larger value – we round down.

If it’s the other way around – we round the number up. This last method is guided by a special rule that we use in case the number is equally distant from the nearest larger and the nearest smaller acceptable value (if we are rounding to the nearest ten, an example for such a case would be the number – which is exactly five units larger than the number and exactly five units smaller than the number ).

Most commonly we use the “round half up” rule, which means that we take the digit we want to round and add the value it represents. For example, if we want to round the number to the nearest ten, we add to it units and, by doing so, round it up to . And if we want to round the number to the nearest thousand, we add hundreds and get the number as the result.

Examples

Natural numbers

Rounding natural numbers is pretty straightforward so we’ll just go through a couple of examples. Let’s say we use the number and we want to get rid of the units by rounding them to the appropriate multiple of ten (ten being the lowest order of magnitude we want to work with).
If we want to round the number up, we will round it to the number . ()

If we want to round the number down, we will round it to the number . ()
If we want to round the number towards zero, we will round it to the number . ()
If we want to round the number away from zero, we will round it to the number . ()
If we want to round the number to the nearest acceptable value (in this case, the nearest multiple of number ), we will have to do a bit of addition and subtraction to decide where we want to go.

Since number is less than nuber , the nearest multiple of to the number is the number , which means that we will round it down to . ()

Integers

The method for rounding integers is a bit different from the method we use to round natural numbers – because what we consider to be up and down when it comes to positive numbers is basically reversed when we are talking about negative numbers. The larger number is the one that is closer to zero, while the smaller number is the one further away from zero.
To make it clearer, let’s look at some more examples. This time we’ll use the number and round it to a multiple of ten.
If we want to round the number up, we will round it to the number . ()
If we want to round the number down, we will round it to the number . ()
If we want to round the number towards zero, we will round it to the number . ()
If we want to round the number away from zero, we will round it to the number . ()

If we want to round the number to the nearest acceptable value (in this case, the nearest multiple of ), we will have to do a bit of addition and subtraction again. Since it doesn’t change the distance between numbers, we will use their positive counterparts (or absolute values) to make it a bit easier for us.

Since number is less than number , the nearest multiple of to the number is the number , which means that we will round it up to . ()

Decimal numbers

Decimal numbers, positive or negative alike, are rounded as integers – the only difference being that decimals can be rounded to the nearest unit or tenth, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths, hundred millionths, billionths etc. – depending on the desired level of precision.

Other than that, the same rules apply.

For example, let’s say we want to round the decimal number to the appropriate tenth.
If we want to round the number up, we will round it to the number . ()
If we want to round the number down, we will round it to the number . ()
If we want to round the number away from zero, we will round it to the number . ()
If we want to round the number to the nearest acceptable value (in this case, the nearest tenth), we have to do a bit of subtraction to decide the direction in which to go in again.

Since number is less than , the nearest tenth to the number is the number , which means that we will round it up to . ()

Fractions

Rounding fractions follows the same principals as rounding decimals. You can convert them to decimal numbers before rounding or you can round them as fractions. In case you want to round them as fractions, you can round them to another fraction or to an integer. In any case, the rules are the same – even if the numbers seem different. Take the fraction you want to round and position it in relation to the fraction you have to round to – just like you would do with decimals or integers. After that, all you have to do is to use the desired method and that is it.

Whether you want to round an natural number, integer, decimal number or fraction, just follow the previous rules and you’ll never have to worry about it again. If you want to practice a bit more or test your knowledge, please feel free to use our worksheets.